© The Authors 2026

1. Introduction

Type III solar radio bursts are among the most intense radio emissions in the Solar System (e.g. Dulk 1985; Reid & Ratcliffe 2014). Decades of high- and low-frequency waveform analysis have revealed crucial insights into the sequence of processes arising during such bursts – from the ejection of coronal beams to the radiation of electromagnetic waves (Ergun et al. 2008; Malaspina & Ergun 2008; Malaspina et al. 2011; Graham & Cairns 2013; Kellogg et al. 2013; Thejappa & MacDowall 2021) – including more recent observations (e.g. Pulupa et al. 2020; Píša et al. 2021; Soucek et al. 2021; Larosa et al. 2022; Jebaraj et al. 2023; Formánek et al. 2025; Pulupa et al. 2025; Ma et al. 2026; Kretzschmar et al. 2026) by the Solar Orbiter (Fox et al. 2016) and Parker Solar Probe (Müller et al. 2020) satellites.

During type III bursts, energetic electron beams generate Langmuir and upper-hybrid wave turbulence in the solar wind, which is subsequently converted into electromagnetic radiation at the fundamental plasma frequency ω_p and its harmonic 2ω_p via a series of wave-wave, wave-particle, and wave-plasma processes. The nonlinear three-wave electrostatic decay (ESD), whereby a Langmuir wave, ℒ, decays into a backscattered wave, ℒ′, and an ion sound wave, 𝒮′, through the channel ℒ → ℒ′+𝒮′ (e.g. Tsytovich 1970; Melrose 1980), plays a central role in the solar wind. Through the generation of backscattered waves, this process enables the emission of harmonic waves at 2ω_p via the three-wave coalescence mechanism ℒ + ℒ′→ℋ (e.g. Melrose et al. 1986; Yoon 2019). Additionally, the nonlinear three-wave electromagnetic decay (EMD) can directly contribute to radio emission at ω_p through the channel ℒ → ℱ + 𝒮, where 𝒮 is an acoustic wave and ℱ is the ordinary electromagnetic wave (e.g. Melrose 1980). In the solar wind, Langmuir waves are quasi-electrostatic and acquire a magnetic component due to the ambient magnetization; hereafter they are referred to as Langmuir/𝒵-mode (ℒ𝒵) waves, which, similarly to Langmuir waves, participate in the processes described above.

On the other hand, density turbulence is ubiquitous in the solar wind and random density fluctuations, δn, of various wavelengths and amplitudes have been measured (e.g. Celnikier et al. 1983, 1987; Kellogg et al. 1999; Krupar et al. 2018, 2020). When the typical wavelength of δn is much larger than that of ℒ𝒵 waves, they can interact with them under some conditions. The strength of these interactions notably depends on the average level of random density fluctuations, ΔN = ⟨(δn/n₀)²⟩^1/2, the electron beam velocity, v_b, and the electron plasma thermal velocity, v_T (n₀ is the average ambient plasma density). When ΔN ≳ 3(v_T/v_b)², linear transformations of Langmuir waves on density fluctuations such as reflection, refraction, tunneling, trapping, or conversion play a crucial role (e.g. Ryutov 1970; Krafft et al. 2013; Voshchepynets et al. 2015). In particular, it was shown that, under such conditions, the linear mode conversion (LMC) at constant frequency is the most efficient process of electromagnetic emission at ω_p in the solar wind (Krasnoselskikh et al. 2019; Krafft & Savoini 2022a; Krafft et al. 2025; Krafft & Volokitin 2025).

Among the various numerical approaches used to study such wave processes, particle-in-cell (PIC) simulations of beam-driven Langmuir wave turbulence have been shown to be a powerful tool (e.g. Rhee et al. 2009; Lee et al. 2019; Krafft & Savoini 2023, and references therein). In this framework, studies have been conducted regarding nonlinear wave-wave processes in homogeneous or inhomogeneous plasmas (Kasaba et al. 2001; Henri et al. 2019; Krafft & Savoini 2021, 2022b; Polanco-Rodríguez et al. 2025a) or aimed at understanding the interactions between wave turbulence and random density fluctuations (Krafft & Savoini 2022a; Krafft et al. 2024, 2025). In a previous work by the authors (Polanco-Rodríguez et al. 2025b), we introduced a new technique using a large number of virtual satellites that record waveforms in a two-dimensional (2D) PIC simulation plane. Unlike global wave diagnostics, which tend to blend all wave phenomena together, this local approach enables the identification of localized wave processes and their temporal evolution in interaction with other mechanisms. Another advantage is that such an approach mimics actual space-recorded waveforms and enables the robust statistical analysis of very large sets of simulation data. By directly comparing simulated waveforms with spacecraft observations in the solar wind (e.g. Polanco-Rodríguez et al. 2026), we can extend our understanding beyond the physical insights derived exclusively from space observations, where many physical quantities cannot be measured simultaneously. Furthermore, in randomly inhomogeneous plasmas, local characterization of wave processes is essential because their behaviour is highly sensitive to density gradients.

This work focuses primarily on studying the mechanisms of ESD and LMC and assessing their relevance under varying physical conditions, with a particular emphasis on two key parameters: the average level of random density fluctuations, ΔN, and the plasma magnetization ratio, ω_c/ω_p, where ω_c is the electron cyclotron frequency. The central objective is to examine the interplay between linear transformations of turbulent waves on random density fluctuations and nonlinear wave-wave interaction processes, using detailed waveform analysis. In this regard, statistical ensembles of waveforms are used to estimate the occurrence rate of the ESD, which is thought to be the most efficient nonlinear wave process in the solar wind under different physical conditions. This approach allows for the study of its competition with LMC and the tracking of the temporal evolution of wave turbulence.

Our results provide new perspectives on the wave processes participating in electromagnetic radiation during type III radio bursts, corroborating earlier results obtained through global analysis techniques. By bridging PIC simulations with experimental observations, this work also establishes a robust foundation for the analysis and interpretation of future space-based waveform data.

2. Numerical simulations

Large-scale and long-term 2D PIC simulations in two spatial and three velocity dimensions are conducted using the SMILEI code (Derouillat et al. 2018). They cover a computational domain (x, y) of size L_x × L_y = 1448²λ_D² (with cell size $\mathrm{\Delta }x=\mathrm{\Delta }y=\sqrt{2}{\lambda }_D$, where λ_D is the electron Debye length) and are conducted up to large times t ≃ 15 000ω_p⁻¹, ensuring that the complete development of wave turbulence is captured. To maintain numerical accuracy over such long timescales and to account for density fluctuations, δn, of only a few percent of the average plasma density n₀, 1800 particles per cell and per species – plasma ions and electrons as well as beam electrons – are employed.

To simulate type III radio bursts’ conditions, a weak and energetic electron beam is injected into the simulation plane (x, y) along the background magnetic field B₀ oriented along the x axis. Initially, the beam electrons are uniformly distributed throughout the entire periodic simulation domain, where they evolve self-consistently. The beam drift velocity, v_b = 12.7v_T ≃ 0.25c, and relative density, n_b = 5 ⋅ 10⁻⁴n₀, are prescribed. The mass and temperature ratios between ions and electrons are m_e/m_i = 1/1836 and T_e/T_i = 10. The cyclotron-to-plasma frequency ratio varies within the range 0 ≤ ω_c/ω_p ≤ 0.14, corresponding to weakly magnetized plasmas such as the solar wind. Some simulations include plasma random density fluctuations, δn, with an average level, ΔN = ⟨(δn/n₀)²⟩^1/2 ≤ 0.05. Initially, they follow a Gaussian distribution in k-space; their wavelengths are much greater than those of beam-driven Langmuir and ℒ𝒵 waves. Then the fluctuations evolve self-consistently during the simulations.

The waveforms are recorded by a set of N = 256 virtual satellites moving through the simulation plane with constant velocity, v_s = |v_s| = 0.3v_T, directed along B₀ (Polanco-Rodríguez et al. 2025b). They are initially distributed uniformly on a grid across the simulation domain. Some of the results presented below are derived from ensemble averages, computed either over the full set of satellites or over selected subsets. The choice of v_s is relevant to solar wind plasmas moving at a speed of around 400 km/s, with the temperature, T_e ≃ 10 eV. The six components of electric and magnetic fields, as well as the density of each species, are recorded. Doppler-shifted frequencies are denoted as $\omega =\overline{\omega }-\mathbf{k}·{\mathbf{v}}_s$, where $\overline{\omega }$ and k are the frequency and wavevector of waves in the plasma (laboratory) frame. As the ion acoustic frequencies are very small in this frame, we can write ${\omega }_{\mathcal{S}}=|{\overline{\omega }}_{\mathcal{S}}-\mathbf{k}·{\mathbf{v}}_s|\simeq |\mathbf{k}·{\mathbf{v}}_s|$. The three-wave resonance condition of the ESD process ℒ → ℒ′+𝒮′ is written as ${\overline{\omega }}_{\mathcal{L}}={\overline{\omega }}_{\mathcal{L}\prime }+{\overline{\omega }}_{\mathcal{S}\prime }$ in the laboratory frame. In the moving satellite frame, it has to be expressed as ω_ℒ = ω_ℒ′ − ω_𝒮′, where the negative sign arises from the Doppler shift.

For additional technical and methodological details, the readers are referred to the previous works (e.g. Krafft & Savoini 2024; Polanco-Rodríguez et al. 2025a). Note that the waveforms shown in Figures 1, 4, 5 and 8 are recorded by a single virtual satellite, whereas the quantities presented in the other figures are obtained by averaging over statistical ensembles of waveforms (each recorded by a different satellite), as specified in the captions.

Fig. 1. Waveforms in a homogeneous and unmagnetized plasma (ΔN = 0, ωc = 0). (a) Time variations in the parallel (grey) and perpendicular (red) electric fields E∥(t) and E⊥(t). (b) Spectral electric field energy |E|2 as a function of the normalized Doppler-shifted frequency ω/ωp and the time ωpt. (c) Time variations in the ion density perturbation δni(t)/n0. (d) Low-frequency spectral energy |δni/n0|2 in the map (ω/ωp, ωpt). (e) High-frequency wave energy spectra |E∥|2 (black) and |E⊥|2 (red), calculated in the time interval 1000 ≲ ωpt ≲ 6000, as a function of ω/ωp. (f) Low-frequency wave energy spectrum |δni/n0|2, in the same time interval and in linear scale, as a function of ω/ωp. (g) Corresponding squared cross-bicoherence $b_c^2$ calculated in the time interval 1000 ≲ ωpt ≲ 6000 for the triplet (E∥, δni, E∥), in the map (ωE∥, ωδni); the extrema bc ≃ 0.72 and bc ≃ 0.68, represented by stars, appear at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })=(0.978,0.0497){\omega }_p$ (first cascade) and $({\omega }_{{\mathcal{L}}^{″}},{\omega }_{{\mathcal{S}}^{″}})=(0.986,0.042){\omega }_p$ (second cascade), respectively. All variables are in arbitrary units.

3. Homogeneous unmagnetized plasma

We start by analysing waveforms from simulations conducted in homogeneous and unmagnetized plasmas. The following two Sections 4 and 5 then explore how plasma density turbulence and magnetization affect the underlying wave phenomena observed in these waveforms. Our focus lies on processes related to electromagnetic wave radiation at ω_p. In this context, two nonlinear processes are of particular interest in homogeneous and unmagnetized plasmas, i.e. the EMD, which produces ordinary electromagnetic waves via a direct channel, and the ESD, which generates ion acoustic waves capable of triggering the EMD process. Note that the coalescence ℒ + 𝒮 → ℱ can also generate ℱ waves, but this process does not play an important role hereafter.

3.1. Electrostatic decay cascades

Figures 1a–d present, in a homogeneous unmagnetized plasma, representative waveforms of the parallel and perpendicular – with respect to the x axis, i.e. the direction of propagation of the beam – electric field components, E_∥(t) and E_⊥(t), (a), the normalized ion density perturbation, δn_i(t)/n₀ = (n_i(t)−n₀)/n₀, (c), as well as the corresponding spectrograms |E(ω, t)|² (b) and |δn_i(ω, t)/n₀|² (d). The time variation in |E(ω, t)|² shows that beam-driven Langmuir waves ℒ are first excited; backscattered ℒ′ waves appear later (ω_pt ≃ 3000), simultaneously with 𝒮′ waves. Furthermore, Figures 1e–f display the energy spectra of Langmuir and ion acoustic waves, calculated within the time interval 1000 ≲ ω_pt ≲ 6000, as a function of the high and low Doppler-shifted frequencies. One observes a double-peaked structure corresponding to beam-driven and backscattered Langmuir waves involved in the ESD process ℒ → ℒ^′ + 𝒮′, together with a smaller peak representing forward propagating Langmuir ℒ″ waves coming from the second decay cascade ℒ′→ℒ^″ + 𝒮^″, as well as two low-frequency peaks representing ion acoustic waves 𝒮′ and 𝒮^″. The wave frequencies measured in the spectra are reported in Table 1.

Then, the three-wave resonance conditions for Doppler-shifted frequencies, i.e. ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{\mathcal{S}\prime }$ (${\overline{\omega }}_{\mathcal{L}}={\overline{\omega }}_{\mathcal{L}\prime }+{\overline{\omega }}_{\mathcal{S}\prime }$ in the immobile plasma frame) and ${\omega }_{{\mathcal{L}}^{\prime }}={\omega }_{{\mathcal{L}}^{″}}+{\omega }_{{\mathcal{S}}^{″}}$, are satisfied with reasonable accuracy. Indeed, $\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|=|{\omega }_{\mathcal{L}}-{\omega }_{\mathcal{L}\prime }|\simeq 0.048{\omega }_p$ and $\mathrm{\Delta }{\omega }_{\mathcal{L}\prime {\mathcal{L}}^{″}}={\omega }_{\mathcal{L}\prime }-{\omega }_{\mathcal{L}″}\simeq 0.04{\omega }_p$. Furthermore, the squared cross-bicoherence b_c², calculated for the triplet (E_∥, δn_i, E_∥) – which is the most representative of the process at play –, exhibits the high values b_c ≃ 0.72 (b_c² = 0.51) and b_c ≃ 0.68 (b_c² = 0.46) for the first and second cascades, respectively (corresponding positions are indicated by stars in Figure 1g). This confirms that the three-wave resonance conditions and phases’ coherence between modes are satisfied, proving the occurrence of two ESD cascades.

The robustness of this result was assessed using surrogate cross-bicoherence data with randomized phases (Theiler et al. 1992). This analysis demonstrates that b_c peaks well above the noise level, which is approximately 0.3. The same test is applied to all cross-bicoherence calculations presented below. In these cases, the noise floor remains unchanged, while its standard deviation is substantially reduced by averaging over multiple waveforms, increasing the statistical significance of b_c.

Furthermore, we can use the measured frequencies ${\omega }_{\mathcal{L}}$, ω_ℒ′, ${\omega }_{\mathcal{L}″}$, ${\omega }_{\mathcal{S}\prime }$ and ${\omega }_{\mathcal{S}″}$ to recover the beam velocity, v_b, the plasma electron temperature, T_e, and the ion acoustic velocity, c_s (and thus the electron-to-ion temperature ratio) by using the following coupled equations derived in the framework of unmagnetized 1D plasma approximation

$$\begin{array}{cc}\hfill \mathrm{\Delta }{\omega }_{{\mathcal{LL}}^{\prime }}& \simeq {\omega }_{\mathcal{L}}-{\omega }_{{\mathcal{L}}^{\prime }}\simeq 2{\omega }_p(1/v_b-c_s/3v_T^2)(c_s-v_s),\hfill \end{array}$$(1)

$$\begin{array}{cc}\hfill \mathrm{\Delta }{\omega }_{{\mathcal{L}}^{\prime }{\mathcal{L}}^{″}}& \simeq {\omega }_{{\mathcal{L}}^{\prime }}-{\omega }_{{\mathcal{L}}^{″}}\simeq 2{\omega }_p(1/v_b-c_s/v_T^2)(c_s+v_s),\hfill \end{array}$$(2)

$$\begin{array}{cc}\hfill \mathrm{\Delta }{\omega }_{{\mathcal{S}}^{\prime }{\mathcal{S}}^{″}}& \simeq 4c_s{v}_s/3v_T^2,\hfill \end{array}$$(3)

where the ion acoustic frequencies, of the order of a few 10⁻³ω_p in the plasma (laboratory) frame, have been neglected. We used ${\omega }_{\mathcal{L}\prime }={\omega }_{{\mathcal{L}}^{″}}+{\omega }_{{\mathcal{S}}^{″}}$ and ${\mathbf{k}}_{\mathcal{L}\prime }={\mathbf{k}}_{{\mathcal{L}}^{″}}+{\mathbf{k}}_{{\mathcal{S}}^{″}}$, as well as the Langmuir and ion acoustic dispersion laws, which lead to $k_{{\mathcal{L}}^{″}}\simeq k_b-2k_0$ and $k_{{\mathcal{S}}^{″}}\simeq -2k_b+3k_0$ in the 1D approximation, where k_b = ω_p/v_b and k₀λ_D = 2c_s/3v_T (e.g. Cairns 1987; Krafft & Savoini 2024); c_s is the ion acoustic velocity. Entering in Equations (1)–(3) the measured frequencies, we get v_b = 13.63v_T, c_s = 0.015v_T, and v_T = 0.017c; these quantities are very close to the simulation parameters, i.e. v_b = 12.7v_T, c_s = 0.026v_T, and v_T = 0.02c. Note that if only one decay cascade occurs, only two equations can be used. Then, in cases in which waves propagate at modest angles relative to the beam direction and in very weakly magnetized solar wind regions near 1 AU, where density turbulence should be weak, high- and low-frequency energy spectra with peaks driven by ESD can be used to diagnose beam and plasma parameters.

3.2. Dynamics of Langmuir wave turbulence

Figure 2 shows the energy spectra ⟨|E_∥|²⟩, ⟨|E_⊥|²⟩, and ⟨|δn_i/n₀|²⟩, averaged over N = 256 waveforms and calculated within two time intervals. In the first time range 1000 ≲ ω_pt ≲ 6000, energy peaks can be identified at frequencies ω_ℒ ≃ 0.98ω_p, ${\omega }_{\mathcal{L}\prime }\simeq 1.025{\omega }_p,$ ${\omega }_{\mathcal{S}\prime }\simeq 0.052{\omega }_p$, and ${\omega }_{{\mathcal{S}}^{″}}\simeq 0.044{\omega }_p$ (Figures 2a,c). As expected, they are very close to those of Figures 1e–f, reported in Table 1. However, note the larger width of the peak of backscattered ℒ′ waves compared to that of the beam-driven ones, due to Doppler-shift effects. One also observes that ⟨|E_⊥|²⟩ < ⟨|E_∥|²⟩ for most frequencies, except near ω ≃ ω_p where ⟨|E_⊥|²⟩ ≃ ⟨|E_∥|²⟩, i.e. in the wavevectors’ region of electromagnetic waves radiated at ω_p.

Fig. 2. High- and low-frequency energy spectra averaged over N = 256 waveforms, as a function of the normalized Doppler-shifted frequency ω/ωp. (a–b) Parallel (black) and perpendicular (red) averaged electric field spectra ⟨|E∥|2⟩ and ⟨|E⊥|2⟩, in the time intervals 1000 ≲ ωpt ≲ 6000 (a) and 6000 ≲ ωpt ≲ 15 000 (b). (c–d) Low-frequency energy spectra ⟨|δni/n0|2⟩, in the same time intervals as (a) and (b), respectively. (a–b): Logarithmic scales. (c–d): Linear scales. All variables are in arbitrary units.

For 6000 ≲ ω_pt ≲ 15 000, all spectral peaks exhibit a significant broadening, attributed to the prolonged occurrence of ESD cascades. Specifically, the second decay cascade becomes evident as a new, smaller peak emerging near ω ≃ 0.99ω_p and corresponding to ℒ^″ waves (Figures 2b,d). Moreover, the electric energy near ω ≃ ω_p is significantly increased, compared to Figure 2a, due to the appearance of Langmuir waves with smaller k ≪ ω_p/v_b involved in the last stages of ESD and of small-k ℱ waves resulting from the EMD ℒ → ℱ + 𝒮 (Krafft et al. 2024). On the other hand, the low-frequency spectrum broadens and extends to both lower and higher frequencies. Ion acoustic waves with ω ≳ 0.06ω_p (ω ≲ 0.04ω_p) are produced by large-k decaying Langmuir waves generated during beam deceleration via ESD (EMD), which are the dominant processes at late times.

3.3. Resonance conditions and phase coherence between waves

To estimate the fraction of waveforms for which three-wave frequency resonance conditions can be met, ${\omega }_{\mathcal{S}\prime }$ is measured as a function of $\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|$ in selected spectra where clear double-peak (ℒ and ℒ^′) structures can be found, as shown in Figure 3a. One can determine that 60% of the analysed spectra (i.e. 150 of 256) reveal the occurrence of three simultaneous high- and low-frequency peaks fulfilling the resonance conditions $|\mathrm{\Delta }{\omega }_{{\mathcal{LL}}^{\prime }}|\simeq |{\omega }_{{\mathcal{L}}^{\prime }}-{\omega }_{\mathcal{L}}|\simeq {\omega }_{\mathcal{S}\prime }$. Moreover, Figures 3b,c present, in the plane (${\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime }$) and for the time interval 1000 ≲ ω_pt ≲ 6000, the squared average cross-bicoherence ⟨b_c⟩² of the triplet (E_∥, δn_i, E_∥), computed by averaging over the 20% of waveforms (out of 256) presenting the highest values of b_c and resonance conditions fulfilled at frequencies where significant spectral energy is observed. The maximum ⟨b_c⟩≃0.67 (⟨b_c⟩² = 0.45) confirms unambiguously that phase coherence between waves is satisfied and thus that ESD indeed occurs for a substantial number of waveforms. Note that no other significant level of cross-bicoherence is found outside the frequency region where ESD manifests, as evidenced in Figure 3b. The dashed line in Figure 3c represents the theoretical curve derived using the ESD resonance condition ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{{\mathcal{S}}^{\prime }}$ and the Langmuir and ion acoustic dispersion relations, leading to the parametric equation $({\omega }_{\mathcal{L}}(k),{\omega }_{{\mathcal{S}}^{\prime }}(2k-k_0))$. A very good agreement between this theoretical curve and the calculated cross-bicoherence maxima is observed.

Fig. 3. (a) Wave frequency distribution in the map ($\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|/{\omega }_p$, ${\omega }_{\mathcal{S}\prime }/{\omega }_p$), obtained by using the 60% of the spectra that are consistent with ESD occurrence (3 peaks identified), out of a set of 256; the dashed line represents the three-wave resonance condition ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{\mathcal{S}\prime }$. (b) Squared cross-bicoherence ⟨bc⟩2 of the triplet (E∥, δni, E∥), in the time interval 1000 ≲ ωpt ≲ 6000, averaged over the 20% of waveforms (out of 256) satisfying the frequency resonance conditions and high phase coherence in a large (${\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime }$) region. (c) Zoom of (b) in the regions 0.95 < ωℒ/ωp < 1.1 and $0.03<{\omega }_{\mathcal{S}\prime }/{\omega }_p<0.1$; the maximum ⟨bc⟩≃0.67 (⟨bc⟩2 = 0.45) is located at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })\simeq (0.98,0.055){\omega }_p$; the dashed line represents the theoretical curve $({\omega }_{\mathcal{L}}(k),{\omega }_{{\mathcal{S}}^{\prime }}(2k-k_0))$ derived using the ESD resonance condition and the wave dispersion relations. All variables are normalized.

In summary, simulations of an initially homogeneous plasma reveal that wavepackets involved in ESD are commonly observed. Our analysis indicates that approximately 20% of the total wavepackets meet the requirements for both three-wave frequency resonance conditions and phase coherence. Additionally, the occurrence of double ESD cascades – while not predominant – is documented, showing that such multistep processes do take place under the conditions of the simulations.

4. Randomly inhomogeneous and unmagnetized plasmas

Density turbulence is ubiquitous in the solar wind. In particular, the transformations of Langmuir and ℒ𝒵 waves excited by electron beams on random density fluctuations of specific wavelength ranges can generate electromagnetic wave radiation at the plasma frequency. Moreover, when the average level of random density fluctuations meets the condition ΔN ≳ 3(v_T/v_b)² (e.g. Ryutov 1970; Krafft et al. 2013), the presence of such inhomogeneities has a crucial impact on the mechanisms leading to electromagnetic wave emission (e.g. Volokitin & Krafft 2018; Krasnoselskikh et al. 2019; Krafft & Savoini 2022a). To go deeper, let us apply the methodology outlined in section 3.

4.1. Impact of density fluctuations

Figure 4 illustrates typical waveforms of parallel and perpendicular electric fields, along with ion density perturbations δn_i(t)/n₀, in plasmas where ΔN ≲ 3(v_T/v_b)² (ΔN = 0) and ΔN ≳ 3(v_T/v_b)² (ΔN = 0.025, 0.05). In the homogeneous plasma case (ΔN = 0, Figure 4a), the waveform exhibits nearly continuous high-amplitude waves, with beat structures lasting ∼100ω_p⁻¹ that intensify at ω_pt ≳ 3000, primarily due to the generation of backscattered Langmuir waves via ESD. In inhomogeneous plasmas (Figures 4b–c), the waveforms show isolated field structures alongside oscillations of significantly reduced amplitudes, tunneling through density humps. Indeed, Langmuir waves undergo reflection, refraction, and trapping in density wells where δn_i < 0 (e.g. Ergun et al. 2008; Krafft et al. 2014; Krafft & Volokitin 2021). For ΔN = 0.025, a wavepacket crosses a density hump (δn_i > 0) at 7000 ≲ ω_pt ≲ 10 000, since its frequency ${\overline{\omega }}_{\mathcal{L}}$ exceeds the local plasma frequency ω_p(1 + δn_i/2n₀). In contrast, for ΔN = 0.05 and in the same time interval, the wavepacket is evanescent when tunneling through a higher density hump where ${\overline{\omega }}_{\mathcal{L}}<{\omega }_p(1+\delta n_i/2n_0)$.

Fig. 4. Waveforms of the parallel and perpendicular electric fields E∥(t) (grey) and E⊥(t) (red), respectively, to which the time variations in the normalized ion density perturbation δni(t)/n0 are superimposed in (b)–(c) (green lines and right axes), for a plasma with different average levels of density fluctuations: ΔN = 0 (a), ΔN = 0.025 (b), and ΔN = 0.05 (c). Electric fields are in arbitrary units.

Although the beats mentioned above are more common in plasmas with ΔN ≲ 3(v_T/v_b)², they can persist even when ΔN ≳ 3(v_T/v_b)², as high amplitude waves are trapped in density wells. In such a case, forward and backward reflected waves can interact linearly near reflection points on density gradients, but also through nonlinear ESD (Krafft et al. 2015; Krafft & Savoini 2024). For ΔN > 0, the most intense wavepackets appear at early times (i.e. 500 ≲ ω_pt ≲ 3000, Figures 4b–c) and gradually fade away, contrary to ΔN = 0, where intense wavepackets are ubiquitous. This attenuation arises as a tail of accelerated beam electrons, formed through wave scattering on density fluctuations, is reabsorbing a significant fraction of Langmuir wave energy, leading to its damping (Krafft & Savoini 2023). This effect is most pronounced for large ΔN = 0.05 (see also Krafft et al. 2024).

4.2. Triggering of electrostatic decay by LMC

Figure 5 presents, in a plasma with ΔN = 0.025, typical waveforms of electromagnetic fields and ion density perturbations, as well as spectral and bicoherence diagnostics. The high- and low-frequency energy spectra |E|² and $|\delta {\stackrel{\sim }{n}}_i/n_0{|}^2$ exhibit four and five main peaks, respectively (Figures 5f,g); $\delta {\stackrel{\sim }{n}}_i(t)=\delta n_i(t)-\delta n(t)$ is the induced ion density perturbation and δn(t) represents the applied density fluctuations that evolve self-consistently. The green vertical lines in Figures 5f,g show that the wave energy peaks correspond to the frequencies listed in Table 2.

Fig. 5. Waveforms in a randomly inhomogeneous plasma with ΔN = 0.025 and ωc = 0. (a) Time variations in the parallel (grey) and perpendicular (red) electric fields E∥(t) and E⊥(t) (left axis), as well as of the superimposed ion density perturbation δni(t)/n0 (blue, right axis). (b) Spectral electric field energy |E|2 in the map (ωpt, ω/ωp). (c) Time variation in the induced ion density perturbation $\delta {\stackrel{\sim }{n}}_i(t)/n_0$ (applied density fluctuations δn(t) have been removed from δni(t) by filtering). (d) Low-frequency spectral energy $|\delta {\stackrel{\sim }{n}}_i/n_0{|}^2$ in the map (ωpt, ω/ωp). (e) Spectral magnetic field energy |B⊥2|2 in the map (ωpt, ω/ωp). (f) High-frequency wave energy spectra |E∥|2 (black) and |E⊥|2 (red) versus ω/ωp, calculated in the time interval $\mathrm{\Delta }T=[1000,5500]{\omega }_p^{-1}$; green labels and vertical lines indicate the Langmuir spectral peaks. (g) Low-frequency wave energy spectrum $|\delta {\stackrel{\sim }{n}}_i/n_0{|}^2$ versus ω/ωp, calculated within ΔT; green labels indicate the ion acoustic waves spectral peaks. (h) Magnetic wave energy spectrum |B⊥2|2 versus ω/ωp, calculated within ΔT. (i) Squared cross-bicoherence $b_c^2$ calculated within ΔT for the triplet $(E_{\Vert },\delta {\stackrel{\sim }{n}}_i,E_{\Vert })$, in the map (${\omega }_{E_{\Vert }},{\omega }_{\delta {\stackrel{\sim }{n}}_i}$)/ωp; bc ≃ 0.69 at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })=(0.975,0.058){\omega }_p$ (first cascade); bc ≃ 0.85 at $({\omega }_{\mathcal{L}″},{\omega }_{\mathcal{S}″})=(0.99,0.047){\omega }_p$ (second cascade); bc ≃ 0.63 at $({\omega }_{\mathcal{L}″},{\omega }_{{\mathcal{S}}^{(3)}})=(0.99,0.033){\omega }_p$ (third cascade); positions in the frequency map are indicated by stars. (j) Squared cross-bicoherence $b_c^2$ calculated within ΔT for the triplet $(B_{\perp 2},\delta {\stackrel{\sim }{n}}_i,E_{\Vert })$, in the map (${\omega }_{B_{\perp 2}},{\omega }_{\delta {\stackrel{\sim }{n}}_i}$)/ωp; bc ≃ 0.85 at (ωℱ, ω𝒮ℱ) = (1.01, 0.04)ωp, as indicated by stars. Parameters are the same as in Figure 1, but with ΔN = 0.025. All variables are in arbitrary units.

Then the resonance conditions ${\omega }_{{\mathcal{L}}^{\prime }}-{\omega }_{\mathcal{L}}={\omega }_{{\mathcal{S}}^{\prime }}\simeq 0.06{\omega }_p,$ ${\omega }_{{\mathcal{L}}^{\prime }}-{\omega }_{{\mathcal{L}}^{″}}={\omega }_{{\mathcal{S}}^{″}}\simeq 0.046{\omega }_p$, and ${\omega }_{{\mathcal{L}}^{(3)}}-{\omega }_{\mathcal{L}″}={\omega }_{{\mathcal{S}}^{(3)}}\simeq 0.027{\omega }_p$ for the three cascades of ESD ℒ → ℒ^′ + 𝒮^′, ℒ^′ → ℒ^″ + 𝒮^″, and ℒ^″ → ℒ⁽³⁾ + 𝒮⁽³⁾ are met with reasonable accuracy (taking into account that the plasma is randomly inhomogeneous). Moreover, the squared cross-bicoherence, b_c², calculated within the time range 1000 ≤ ω_pt ≤ 5500 (Figure 5i) for the triplet $(E_{\Vert },\delta {\stackrel{\sim }{n}}_i,E_{\Vert })$ reaches, around the frequencies listed above, high values for the first (b_c ≃ 0.69), the second (b_c ≃ 0.85), and the third (b_c ≃ 0.63) decay cascades, at positions indicated by stars (b_c² ≃ 0.48, 0.72, and 0.4, respectively), confirming the occurrence of three ESD cascades. As shown by the waveforms of electric fields and ion density perturbations and their corresponding spectral energies (Figures 5a–d), these cascades occur at the time when beam-driven Langmuir waves reach a density gradient and interact with the reflected backscattered waves, ℒ^′. Due to their large amplitudes, ℒ and ℒ′ waves can trigger ESD, generating ion acoustic waves at ${\omega }_{\mathcal{S}\prime }$ (Figures 5c–d,i). Meanwhile, ℒ″ and ℒ⁽³⁾ waves are excited at frequencies ${\omega }_{{\mathcal{L}}^{″}}\simeq 0.99{\omega }_p$ and ${\omega }_{{\mathcal{L}}^{(3)}}\simeq 1.017{\omega }_p$ within the range 2800 ≲ ω_pt ≲ 3800 (Figures 5b,f). This shows that wave reflections on random density fluctuations can locally trigger multiple ESD cascades.

On the other hand, the appearance of large amplitude magnetic energy at ω ≃ ω_p within 1800 ≲ ω_pt ≲ 2500 is the signature of the linear mode conversion process LMC at constant frequency of Langmuir waves ℒ into fundamental ordinary electromagnetic waves, hereafter referred to as ℱ waves. This process has been shown to be the most efficient in generating ℱ waves in a plasma with ΔN ≳ 3(v_T/v_b)² (Krafft & Savoini 2022a; Krafft et al. 2024, 2025). Furthermore, LMC of Langmuir waves on density fluctuations can trigger the EMD process, ℒ → ℱ + 𝒮_ℱ, where 𝒮_ℱ is an ion acoustic wave, as a result of the simultaneous excitation of ℒ and ℱ waves at significant amplitudes through scattering on δn and conversion, respectively. This nonlinear process can then occur at much earlier times than in a homogeneous plasma. It is likely responsible for the magnetic energy |B_⊥2|² observed at ω ≳ ω_p within 2500 ≲ ω_pt ≲ 3500 (Figure 5e). Recall that, in an unmagnetized plasma, the field component B_⊥2 = B_z (perpendicular to the two-dimensional simulation plane (x, y)) dominates the magnetic energy, which is primarily carried by ordinary wave modes. The cross-bicoherence calculated in Figure 5j using the triplet $(B_{\perp 2},\delta {\stackrel{\sim }{n}}_i,E_{\Vert })$ – which is the most representative of the process at play – shows that the maximum b_c ≃ 0.85 (b_c² ≃ 0.72) is reached at ${\omega }_{{\mathcal{S}}_{\mathcal{F}}}\simeq 0.04{\omega }_p$ and ${\omega }_{\mathcal{F}}\simeq 1.01{\omega }_p$ (see star). This confirms that the electromagnetic waves emitted at ω ≳ ω_p originate from the EMD of the backscattered wavepacket through the channel ℒ′ → ℱ + 𝒮_ℱ, with frequency ${\omega }_{{\mathcal{L}}^{\prime }}\simeq 1.045{\omega }_p$. This result is consistent with the earlier findings of the authors (Krafft et al. 2024) which were derived using a global approach (as opposed to the local method employed here).

The magnetic signature observed at ω ≲ ω_p, simultaneously with EMD, is likely to correspond to ℱ waves generated by LMC near density gradients where ℒ waves are reflected (ω_pt ≃ 2800). This conversion process can stimulate EMD. Similarly, as previously discussed, linear transformations of ℒ waves on density fluctuations can trigger ESD as well, by generating backscattered ℒ′ waves of significant amplitudes. Then, through the growth of 𝒮′ waves with wavevectors close to those of ion acoustic waves 𝒮_ℱ involved in the EMD mechanism, it can, in turn, also be stimulated (Krafft et al. 2024). This kind of electromagnetic wave generation following the occurrence of LMC is frequently observed in the waveforms recorded in our simulations with randomly inhomogeneous plasmas, which confirms that LMC can trigger nonlinear phenomena at early times, as suggested previously by the authors.

4.3. Occurrence of electrostatic decay in plasmas with density turbulence

Figures 6a,b display PDFs of wave frequencies in the map ($\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|/{\omega }_p$, ${\omega }_{\mathcal{S}\prime }/{\omega }_p$), obtained using selected high-frequency spectra featuring a double-peak structure, in plasmas with ΔN = 0.025 (a) and ΔN = 0.05 (b). The distributions are scattered along the ESD frequency resonance condition $|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }|=|{\omega }_{\mathcal{L}\prime }-{\omega }_{\mathcal{L}}|={\omega }_{\mathcal{S}\prime }$ (represented by dashed lines), which is hardly satisfied due to wave scattering on density fluctuations, but not locally prohibited (Krafft & Savoini 2024). More precisely, around 20% of the spectral peaks satisfy the frequency resonance condition. Compared to the homogeneous plasma case, the distributions are noticeably shifted towards larger ion acoustic frequencies, likely due to the presence of scattered Langmuir waves with larger k. Additionally, the frequency band $\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|$ widens with increasing ΔN, consistent with the frequency broadening ∼ΔNω_p induced by density fluctuations and indicating a smaller number of waves satisfying resonance conditions.

Fig. 6. (a–b) PDFs of wave frequencies in the plane ($\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|/{\omega }_p$, ${\omega }_{\mathcal{S}\prime }/{\omega }_p$), obtained using the 20% of spectra that show a double-peak structure and satisfy the frequency resonance condition, out of a set of 256, for ΔN = 0.025 and ΔN = 0.05, respectively; the dashed lines represent the three-wave resonance condition ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{{\mathcal{S}}^{\prime }}$. (c–d) Squared average cross-bicoherence ⟨bc⟩2 of the triplet ($E_{\Vert },\delta {\stackrel{\sim }{n}}_i,E_{\Vert }$), calculated in the time interval 1000 ≲ ωpt ≲ 6000, in the frequency map $({\omega }_{\mathcal{L}}/{\omega }_p,{\omega }_{\mathcal{S}\prime }/{\omega }_p)$, for ΔN = 0.025 (⟨bc⟩≃0.54 at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })\simeq (0.98,0.055){\omega }_p$) and ΔN = 0.05 (⟨bc⟩≃0.6 at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })\simeq (0.97,0.065){\omega }_p$), respectively. Averaging is done over the 8% (c) and 5% (d) of spectra with waves satisfying resonance conditions at frequencies corresponding to high levels of cross-bicoherence, out of 256. The dashed lines represent the parametric curves $({\omega }_{\mathcal{L}}(k)+\alpha \mathrm{\Delta }N{\omega }_p/2,{\omega }_{\mathcal{S}\prime }(2k-k_0))$ for α = −1, 1, derived using the resonance condition ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{\mathcal{S}\prime }$ and the wave dispersion relations. Simulation parameters are the same as in Figure 1, but with ΔN > 0. All variables are normalized.

Figures 6c,d present the squared average cross-bicoherence ⟨b_c⟩². Averaging is performed over the 5% (8%) of waveforms with high levels of b_c and satisfying frequency resonance conditions, observed at wave energy maxima for ΔN = 0.05 (ΔN = 0.025). The bicoherence maxima are bounded by the parametric curves $({\omega }_{\mathcal{L}}(k)+\alpha \mathrm{\Delta }N{\omega }_p/2,{\omega }_{\mathcal{S}\prime }(2k-k_0))$ with α = −1, 1 (dashed lines). The extrema ⟨b_c⟩≃0.54 (⟨b_c⟩² ≃ 0.29) and ⟨b_c⟩≃0.6 (⟨b_c⟩² ≃ 0.36) are found for ΔN = 0.025 (c) and ΔN = 0.05 (d), respectively. Once again, one observes that the frequencies of ion acoustic and Langmuir waves are widespread, due to the significant wave scattering on density fluctuations. The highest levels of bicoherence correspond roughly to similar frequencies $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })$ for both ΔN, but not exactly. Indeed, ESD occurs only in localized plasma regions where the density turbulence is weak enough to preserve the phase coherence between waves, which depends on ΔN. Finally, note that ⟨b_c⟩≃0.5 (⟨b_c⟩² ≃ 0.25) at $({\omega }_{\mathcal{F}},{\omega }_{{\mathcal{S}}_{\mathcal{F}}})\simeq (1.03,0.016){\omega }_p$ (Figures 6c,d), suggesting the occurrence of EMD, stimulated by LMC.

4.4. Spectral characteristics of Langmuir wave turbulence

Figures 7a–f show the high- and low-frequency averaged spectra ⟨|E_∥|²⟩, ⟨|E_⊥|²⟩, and $⟨|\delta {\stackrel{\sim }{n}}_i/n_0{|}^2⟩$ versus ω/ω_p, for ΔN = 0, 0.025 and 0.05, in the time ranges 1000 ≲ ω_pt ≲ 6000 (a–c) and 6000 ≲ ω_pt ≲ 15, 000 (d–f). At early times (ω_pt ≲ 6000), when ΔN increases from 0 to 0.05 (Figures 7a-c), the energy ⟨|E_∥|²⟩ of beam-driven ℒ (backscattered ℒ^′) waves decreases (tends to a quasi-flat scattered distribution), as a result of wave scattering on density inhomogeneities and damping by beam reabsorption. Random density fluctuations are responsible for the broadening of ⟨|E_∥|²⟩ and ⟨|E_⊥|²⟩, which increases with ΔN. For ΔN = 0, the significant increase in their amplitudes with time within the frequency range 0.99 ≲ ω/ω_p ≲ 1.01 results from Langmuir wave energy transport to small k through nonlinear processes such as ESD and EMD.

Fig. 7. Energy spectra averaged over 256 waveforms in the time range ΔT, as a function of the Doppler-shifted frequencies ω/ωp, for ΔN = 0 (blue), ΔN = 0.025 (black), and ΔN = 0.05 (red). (a–c) Parallel ⟨|E∥|2⟩ and perpendicular ⟨|E⊥|2⟩ electric field spectra, as well as ion acoustic energy spectrum $\langle |\delta {\stackrel{\sim }{n}}_i/n_0{|}^2\rangle$, for ΔT = [1000, 6000]ωp−1. (d–f) ⟨|E∥|2⟩, ⟨|E⊥|2⟩, and $\langle |\delta {\stackrel{\sim }{n}}_i/n_0{|}^2\rangle$, for ΔT = [6000, 15 000]ωp−1. All variables are in arbitrary units.

For ΔN > 0, $\langle |\delta {\stackrel{\sim }{n}}_i/n_0{|}^2\rangle$ is strongly flattened in both early and late time intervals, showing the predominance of Langmuir wave transformations on density fluctuations and a lower occurrence rate of three-wave interaction processes as ESD and EMD. The extension of $\langle |\delta {\stackrel{\sim }{n}}_i/n_0{|}^2\rangle$ to ω ≲ 0.02ω_p is explained by (i) the occurrence of EMD stimulated at early stages by LMC, producing ion acoustic waves of frequencies lower than those generated by ESD, and (ii) the fact that ℒ waves scattered on δn produce, through ESD and EMD, oblique ion acoustic waves with smaller frequencies ω(k)≃|kv_s cos θ|. Moreover, due to Langmuir wave scattering and its impact on three-wave resonance conditions, the distribution $\langle |\delta {\stackrel{\sim }{n}}_i/n_0{|}^2\rangle$ extends over a frequency range larger than for ΔN = 0 (compare with Figure 2). Note that, while high levels of cross-bicoherence are reached for 0.04 ≲ ω/ω_p ≲ 0.07 (Figures 6c,d), $⟨|\delta {\stackrel{\sim }{n}}_i/n_0{|}^2⟩$ peaks at ω ≃ 0.02ω_p. This can be attributed to two main factors. First, the frequencies of ion acoustic waves involved in ESD are broadly distributed (Figures 6a–d); second, most of the analysed waveforms lack a very distinct ESD signature. As a result, when averaged, these waves contribute only limited statistical significance to the ESD process.

In addition, Figures 7d–f show the corresponding distributions within a later time interval 6000 ≲ ω_pt ≲ 15, 000. We observe that all spectra are significantly broadened, due to ESD and EMD processes for ΔN = 0, and to transformations of Langmuir waves on δn for ΔN > 0. For ΔN > 0 (ΔN = 0), both electric energy spectra show that the Langmuir wave turbulence is strongly (weakly) damped, due to energy reabsorption by the beam (Krafft & Savoini 2023). When ΔN > 0, the distributions’ shapes differ significantly from those observed earlier (Figure 7c). Indeed, at ω_pt ≳ 6000, Langmuir waves are already substantially damped, and intense wavepackets become rare, as well as wave-wave interactions.

In summary, our analysis demonstrates that ESD persists in localized regions of randomly inhomogeneous plasmas. Around 20% of the 256 spectra are consistent with frequency resonance conditions, while 8% only retain also phase coherence between wave triplets. However, the Langmuir and ion acoustic wave spectra appear to be more scattered and diffuse, with diminished peak intensities at high frequencies. This shows that linear transformations of Langmuir waves on density fluctuations dominate, particularly through the LMC process, which efficiently converts Langmuir waves into electromagnetic waves at constant frequency. Despite reduced phase coherence, decay processes involving multiple cascades endure in inhomogeneous plasmas. In particular, we provide clear evidence that LMC stimulates both EMD and ESD. By adopting a local approach, this work corroborates and extends findings previously identified or suggested through global analysis in Krafft et al. (2024), offering complementary insights into the interplay between nonlinear interactions between waves and linear transformations of waves in randomly inhomogeneous plasmas.

5. Weakly magnetized homogeneous plasmas

In magnetized plasmas, the beam-plasma instability generates upper-hybrid wave turbulence. The excited quasi-electrostatic waves are also called magnetized Langmuir or Langmuir/𝒵-mode waves, and are below referred to as ℒ𝒵 waves. In weakly magnetized, homogeneous plasmas, the dynamics of turbulent ℒ𝒵 waves is primarily governed by nonlinear wave-wave interactions; the influence of plasma magnetization on these processes is examined below.

5.1. Impact of plasma magnetization on wave processes

In a weakly magnetized plasma (ω_p ≫ ω_c), ℒ𝒵 waves follow at small k the dispersion of the slow-extraordinary mode, the so-called electromagnetic 𝒵-mode with the cutoff frequency ω ≃ ω_p − ω_c/2. The transition between Langmuir-like and the 𝒵-mode-like wave dispersions occurs along the parallel direction around the wavenumber k_*λ_D = (v_T/c)(1 + ω_p/ω_c)^−1/2 (Cairns & Layden 2018). Other electromagnetic modes are emitted at ω ∼ ω_p, i.e. the ordinary 𝒪-mode and the fast-extraordinary 𝒳-mode, with the cutoff frequencies ω = ω_p and ω ≃ ω_p + ω_c/2, respectively.

Figures 8a–e show representative waveforms of the parallel and perpendicular electric fields, together with the ion density perturbation and the corresponding spectrograms of |E|², |B|² and |δn_i/n₀|². At early times (ω_pt ≲ 4000), the parallel field component E_∥(t) dominates over the perpendicular component (Figure 8a). As time progresses, the amplitude of E_⊥(t) increases and wave beatings emerge around ω_pt ≳ 3500, together with backscattered ℒ𝒵′ and ion acoustic 𝒮′ waves (Figures 8b–d). Jointly, magnetic energy appears at ω ≳ ω_p, indicating the generation of ℒ𝒵 and/or 𝒪-mode waves produced through electrostatic and electromagnetic decay, respectively. Later (ω_pt ≳ 8500), 𝒵-mode waves are generated at frequencies ω ≲ ω_p during the final stage of ESD (Polanco-Rodríguez et al. 2025a).

Fig. 8. Waveforms in a homogeneous and weakly magnetized plasma with ωc/ωp = 0.07. (a) Time variations in the parallel (grey) and perpendicular (red) electric fields E∥(t) and E⊥(t). (b) Spectral electric field energy |E|2 in the map (ωpt, ω/ωp). (c) Time variation in the ion density perturbations δni(t)/n0. (d) Low-frequency spectral energy |δni/n0|2 in the map (ωpt, ω/ωp). (e) Spectral magnetic field energy |B|2 in the map (ωpt, ω/ωp). (f) High-frequency wave energy spectra |E∥|2 (black) and |E⊥|2 (red), versus ω/ωp, calculated in the time interval $\mathrm{\Delta }T=[0,6500]{\omega }_p^{-1}$; the green vertical lines indicate the spectral peaks of excited ℒ𝒵 waves. (g) Low-frequency wave energy spectrum |δni/n0|2 versus ω/ωp, calculated within ΔT; the green vertical lines indicate the spectral peaks of excited ion acoustic waves. (h) Squared cross-bicoherence $b_c^2$ calculated within ΔT for the triplet (E∥, δni, E∥), in the map (ωE∥/ωp, ωδni/ωp); bc ≃ 0.6 at $({\omega }_{\mathcal{LZ}},{\omega }_{\mathcal{S}\prime })=(0.979,0.059){\omega }_p$ (first cascade); bc ≃ 0.63 at $({\omega }_{\mathcal{LZ}″},{\omega }_{\mathcal{S}″})=(0.987,0.048){\omega }_p$ (second cascade); positions are indicated in the map by stars. All variables are in arbitrary units.

The energy spectra of ℒ𝒵 and ion acoustic waves calculated in the range ω_pt ≲ 6500 are shown in Figures 8f–g as a function of ω/ω_p. One observes the signatures of the two decay cascades ℒ𝒵 → ℒ𝒵^′ + 𝒮^′ and ℒ𝒵′→ℒ𝒵^″ + 𝒮^″ in the form of spectral peaks with frequencies corresponding to ℒ𝒵, ℒ𝒵′, ℒ𝒵^″, 𝒮′, and 𝒮^″ waves, which are listed in Table 3.

The three-wave resonance conditions ${\omega }_{\mathcal{LZ}}={\omega }_{\mathcal{LZ}\prime }-{\omega }_{\mathcal{S}\prime }$ and ${\omega }_{\mathcal{LZ}\prime }={\omega }_{\mathcal{LZ}″}+{\omega }_{\mathcal{S}″}$ are satisfied with good accuracy. Furthermore, the squared cross-bicoherence b_c² calculated within the same time interval for the triplet (E_∥, δn_i, E_∥) provides that b_c ≃ 0.6 at the frequencies corresponding to the first and second cascades (see Figure 8h and its caption).

Figures 8a–h confirm the occurrence of decay in the recorded waveforms. Moreover, the late increase in E_⊥(t) in Figure 8a results from ESD transporting wave energy to smaller, more oblique wavevectors (Polanco-Rodríguez et al. 2025b). At ω_pt ≳ 8500, 𝒵-mode waves appear during the last stage of ESD (Polanco-Rodríguez et al. 2025a), when ion acoustic waves’ frequencies satisfy ω ≲ 0.035ω_p. However, clear cross-bicoherence can hardly be evidenced at this time, due to the rarity of phase-coherent wavepackets in the developed wave turbulence. 𝒪-mode waves are generated at ω_pt ≳ 5000 via EMD, with low-frequency signatures at ${\omega }_{\mathcal{S}}\simeq 0.025{\omega }_p$. Note that the generation of electromagnetic waves via the EMD process in a weakly magnetized plasma will be thoroughly treated in a forthcoming paper.

5.2. Magnetic signatures of decaying ℒ𝒵 waves

Figures 9a,b show PDFs of the wave frequencies in the map ($|\mathrm{\Delta }{\omega }_{{\mathcal{LL}}^{\prime }}|/{\omega }_p,{\omega }_{\mathcal{S}\prime }/{\omega }_p)$, for 60% (a) and 58% (b) of the N = 256 spectra that feature a double-peak structure, in magnetized plasmas with ω_c/ω_p = 0.005 and 0.07, respectively. Most of these spectra are on the resonance condition curve $|\mathrm{\Delta }{\omega }_{{\mathcal{LL}}^{\prime }}|\simeq {\omega }_{\mathcal{S}\prime }$. The corresponding squared average cross-bicoherence ⟨b_c⟩² is presented in Figures 9c–f, for the triplets (E_∥, δn_i, E_∥) (c,d) and (B_∥, δn_i, E_∥) (e,f), respectively, within the time domain 1000 ≲ ω_pt ≲ 6000. For the first triplet, high cross-bicoherence levels reach ⟨b_c⟩≃0.67 (⟨b_c⟩² ≃ 0.45) in panel (d) and are aligned along the theoretical curves representing the ESD three-wave resonance conditions in a weakly magnetized plasma (compare Figures 3c and 9c,d). The proportion of waveforms (out of 256) exhibiting clear ESD signatures (resonance conditions together with phase coherence) is 23% (c) and 25% (d). This closely matches the unmagnetized plasma case. As ESD occurs for ω_c = 0 and ω_c > 0 at close frequencies and wavevectors, similar domains in planes (ω_E∥, ω_δni) and (ω_B∥, ω_δni) present high values of ⟨b_c⟩². This shows that the efficiency of ESD at large k is only slightly affected by weak plasma magnetization. This was already studied in a previous work using a global approach (Polanco-Rodríguez et al. 2025a); however, the methodology presented here can be applied to analyse the actual ℒ𝒵 waveforms observed by satellites in the solar wind.

Fig. 9. (a–b) Wave frequency distributions in the plane ($\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|/{\omega }_p$, ${\omega }_{\mathcal{S}\prime }/{\omega }_p$), for Ns selected spectra out of a set of N = 256; (a) : Ns/N≃60%, ωc/ωp = 0.005; (b) : Ns/N≃58%, ωc/ωp = 0.07; both distributions are calculated within the time interval ΔT = [1000, 6000]ωp−1. The dashed lines represent the ESD resonance condition $\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|={\omega }_{\mathcal{LZ}\prime }-{\omega }_{\mathcal{LZ}}={\omega }_{\mathcal{S}\prime }$. (c–d) Square cross-bicoherence ⟨bc⟩2 averaged over Ns selected spectra with both high bc and ESD frequency resonance conditions, within ΔT, in the map (${\omega }_{\mathcal{LZ}}/{\omega }_p,{\omega }_{\mathcal{S}\prime }/{\omega }_p$), for the triplet (E∥, δni, E∥): Ns/N ≃ 23% (c) and Ns/N ≃ 25% (d). (e–f) Square cross-bicoherence ⟨bc⟩2 averaged over Ns selected spectra, within ΔT, in the map (${\omega }_{\mathcal{LZ}}/{\omega }_p,{\omega }_{\mathcal{S}\prime }/{\omega }_p$), for the triplet (B∥, δni, E∥) : Ns/N ≃ 3.5% (e) and Ns/N ≃ 13% (f). (c–f) : the dashed lines represent the theoretical curves described in Figure 3. (Upper row) : ωc/ωp = 0.005. (Bottom row) : ωc/ωp = 0.07.

As ℒ𝒵 waves are weakly (strongly) magnetized at large (small) k-scales (see also Figures 9e,f), they generate through ESD waves with substantial magnetic components. This conclusion is also valid for the higher order ESD cascades, which are, however, not relevant at early times ω_pt ≲ 6000. As plasma magnetization decreases, the spectral region of waves with large magnetic signatures – around k_*– is also reduced, and larger times are needed for decay to reach such a small-k region. This explains why fewer waveforms presenting both resonance conditions and high cross-bicoherence level are observed for the triplet (B_∥, δn_i, E_∥) at ω_c/ω_p = 0.005 (3.5% in Figure 9e) compared to ω_c/ω_p = 0.07 (13% in Figure 9f).

5.3. Wave turbulence spectra versus magnetization

Figures 10a–f show the spectral distributions of ⟨|E_∥|²⟩, ⟨|E_⊥|²⟩, and ⟨|δn_i/n₀|²⟩, averaged over 256 waveforms, in the time ranges 1000 ≲ ω_pt ≲ 6000 (a–c) and 6000 ≲ ω_pt ≲ 15 000 (e–f), for different magnetization ratios. When ω_c/ω_p increases, both electric energies decrease within the frequency range 0.99ω_p ≲ ω ≲ 1.02ω_p (Figures 10a,d), confirming the impact of magnetization on ℒ𝒵 waves of small k (i.e. at ω ∼ ω_p). Indeed, ℒ𝒵 wave energy cannot penetrate the region surrounding k ∼ 0 when ω_c > 0 (Polanco-Rodríguez et al. 2025a). In this frequency range only, and mostly at advanced times, ⟨|E_⊥|²⟩ significantly exceeds ⟨|E_∥|²⟩ when ω_c/ω_p ≤ 0.14, so that the perpendicular energy increases at small k-scales, when ℒ𝒵 waves become quasi-electromagnetic (see also Polanco-Rodríguez et al. 2025b). On the other hand, the electric energy spectra broaden significantly over time (compare Figures 10a–b and 10d–e), due to the beam that excites ℒ𝒵 waves of larger k during its relaxation and the redistribution of energy over smaller k-scales by ESD cascades. In particular, a small shift towards higher frequencies can be observed at ω ∼ 1.03ω_p for ω_c/ω_p ≃ 0.14, which is also visible at ω ∼ 0.055ω_p for the ion acoustic waves produced by ESD (Figures 10c,f). Note that Krauss-Varban (1989) calculated the linear growth rate of ℒ𝒵 waves excited by a beam and also observed a slight k-shift with increasing ω_c/ω_p. For ω_c/ω_p < 0.14, the low-frequency spectral features are comparable to those of Figure 2, showing the weak impact of magnetization. The time evolution of spectra at any ω_c/ω_p shows the broadening of ⟨|δn_i/n₀|²⟩ and its extension towards higher (lower) frequencies over time, due to beam relaxation (higher order decay cascades).

Fig. 10. Energy spectra averaged over N = 256 waveforms, as a function of ω/ωp, for different magnetization ratios ωc/ωp = 0, 0.02, 0.07, and 0.14 (see the legend), in the time ranges 1000 ≲ ωpt ≲ 6000 (top row) and 6000 ≲ ωpt ≲ 15 000 (bottom row). (a,d) Parallel electric energy ⟨|E∥|2⟩. (b,e) Perpendicular electric energy ⟨|E⊥|2⟩. (c,f) Ion acoustic wave spectrum ⟨|δni/n0|2⟩. (a–b, d–e): Logarithmic scales. (c,f): Linear scales. All variables are in arbitrary units.

6. Discussion and conclusion

Our study investigates wave processes occurring during type III solar radio bursts by using large-scale and long-term PIC simulations in two spatial and three velocity dimensions. By reproducing waveforms that closely match spacecraft observations in the solar wind, we can identify the main nonlinear and linear wave phenomena. Notably, we observe wave-wave interaction processes such as ESD and EMD, alongside linear transformations of turbulent electrostatic wavepackets on random density fluctuations, including reflection, refraction, trapping, tunneling, and, crucially, LMC at constant frequency. A key finding is the dynamic interplay and competition between these processes. Specifically, we demonstrate that wave scattering and subsequent LMC, which strongly shape the evolution of the ℒ𝒵 wave turbulence, can initiate nonlinear processes much earlier than expected in plasmas without random density fluctuations.

6.1. Reliability of the study

The robustness and scope of this study depend on several key aspects. First, the results obtained here remain valid over a broad range of beam and plasma parameters: beam density, velocity, and temperature, as well as plasma magnetization, average level of density fluctuations, electron and ion temperatures, and of satellite velocity. Second, we used high-frequency resolution measurements and employed multiple diagnostics, including cross-bicoherence (validated by the surrogate method) and statistical analyses. Third, the study is performed within a 2D plasma framework, in which all six field components, as well as ion and electron densities and full 3D particle velocities, are resolved at every time step. Despite the reduced spatial dimensionality, most of the results are expected to remain qualitatively valid in 3D plasmas. These aspects are discussed in more detail below.

The simulation parameters used in this work are representative of solar wind plasmas between 0.07 and 1 au. In particular, the average level of density fluctuations, ΔN ≤ 0.05, and the magnetization ratio, ω_c/ω_p ≤ 0.14, span the range of conditions typically encountered in these regions. The chosen beam velocity, v_b ≃ 0.25c, is characteristic of type III electron beams. Variations in v_b would primarily affect the number of ESD cascades and their onset time. As the beam naturally slows down due to relaxation associated with wave emission (Krafft & Savoini 2023), it can access a significant range of velocities below 0.25c – consistent with the beam velocities measured in the solar wind during type III solar bursts (Krupar et al. 2015) – and therefore excite a wide range of Langmuir and ℒ𝒵 wavelengths. Regarding beam density, we adopt n_b/n₀ = 5 ⋅ 10⁻⁴, which is the most realistic value currently achieved in 2D PIC simulations. Type III beams are typically 10 to 50 times less dense; however, this value represents to date the lowest beam density used in such large-scale PIC simulations. Increasing n_b/n₀ beyond 10⁻³ would drive the system out of the kinetic regime required to model electromagnetic radiation processes by beam-generated Langmuir and ℒ𝒵 wave turbulence. In contrast, using lower beam density ratios becomes computationally prohibitive, as the growth rate of the beam–plasma instability scales with n_b. Finally, the beam temperature ratio, taken here as v_Tb/v_b ≃ 0.1, is typical of Type III electron beams. Increasing this ratio beyond ∼0.3 would likely suppress the beam–plasma instability and the associated wave processes, while decreasing it below ∼0.05 would lead to the unrealistic cold beam regime.

The influence of ion and electron temperatures on the ESD process has been investigated using 2D PIC simulations in Krafft & Savoini (2024). When T_e/T_i ≲ 3, ion acoustic waves become damped and behave as quasi-modes, while the ESD process is progressively replaced by nonlinear induced scattering on thermal ions (NLIS). To properly capture ESD dynamics, we therefore adopt T_e/T_i = 10, a value consistent with certain solar wind conditions and representative of the range 3 ≲ T_e/T_i ≲ 10 where the qualitative properties of ESD are not significantly altered. In addition, reducing the electron temperature from 200 to 20 eV has been shown to lead to a slower and weaker development of the ESD process, although its underlying kinematics and overall dynamics remain qualitatively unchanged (Polanco-Rodríguez et al. 2025a). For this reason, both the temperature ratio and the electron temperature are fixed to values that ensure a sufficiently rapid onset and appreciable amplitude of the waves involved in ESD, thereby enabling a meaningful study of its competition with other wave processes.

A 2D geometry, which retains both the parallel and perpendicular components (with respect to the ambient magnetic field) of fields and wavevectors, already provides a sufficient framework for the development and evolution of the wave processes considered here. Moreover, we have investigated the LMC in both 2D and 3D geometries and found that the main qualitative features of electromagnetic radiation at ω_p via LMC are preserved in both cases (Krafft & Volokitin 2025; Krafft et al. 2025). These include, for instance, the dominance of 𝒵-mode wave energy over 𝒪- and 𝒳-mode energies, as well as the absence of 𝒳-mode waves under specific conditions of magnetization and density turbulence. Although some quantitative differences do arise – such as scaling the rate of growth of the electromagnetic energy with the normalized thermal velocity v_T/c, which follows different power-laws in 2D and in 3D geometries, for example – these effects do not alter the overall qualitative features of the processes considered here.

In addition, we have recently shown very good agreement between 3D Solar Orbiter observations of ESD to the 𝒵-mode and results of 2D PIC simulations performed under closely matched conditions (Polanco-Rodríguez et al. 2026). Similarly, the results of 2D PIC simulations on electromagnetic wave radiation at ω_p via LMC are consistent with 3D analytical calculations based on weak turbulence theory extended to randomly inhomogeneous plasmas (Krafft & Volokitin 2025; Krafft et al. 2025). Both approaches are also consistent with observations from the Parker Solar Probe and the Solar Orbiter satellites (Pulupa et al. 2025; Kretzschmar et al. 2026), which detected only 𝒪-mode waves propagating away from the source regions. Taken together, these results indicate that the conclusions drawn in this work from the 2D geometry should also remain relevant in 3D configurations, at least from a qualitative standpoint.

6.2. From simulations to observations

Electrostatic decay (ESD) is widely regarded as the most efficient nonlinear wave-wave interaction process in the solar wind and is a cornerstone in understanding type III solar radio bursts. Although EMD and LMC have yet to be unambiguously observed in the solar wind, simultaneous detections of Langmuir and acoustic waves consistent with the ESD mechanism have been reported in previous studies (Henri et al. 2009; Malaspina et al. 2011; Graham & Cairns 2013; Kellogg et al. 2013). However, these have not provided definitive evidence for the occurrence of ESD, due to low statistics and/or a lack of significant phase coherence between waves.

We advance this approach by statistically analysing a large dataset of PIC-simulated waveforms under a controlled framework. In addition to frequency spectra and resonance conditions, our study integrates cross-bicoherence diagnostics and analysis of magnetic energy evolution, ion density waveforms under varying levels of density turbulence, and 3D electric fields. This framework not only aligns PIC simulation results with solar wind observations but, being more comprehensive than space-recorded waveforms, also demonstrates the stimulation of nonlinear processes by LMC and enables evidence of ESD occurrence under different plasma conditions. In homogeneous plasmas (randomly inhomogeneous plasmas with ΔN ≳ 3(v_T/v_b)²), our analysis reveals three-wave ESD resonance conditions in 60% (20%) of the simulated waveforms, and only in 20% (8%) of cases when the phase coherence condition between waves is also fulfilled. This shows that ESD occurs frequently in homogeneous plasmas and, despite strong scattering and LMC, is not fully prohibited in plasmas where density fluctuations strongly interact with Langmuir waves.

Despite some discrepancies, our findings align with space-based waveform analyses. Graham & Cairns (2013) reported that about 40% of the analysed waveforms exhibit spectral peaks – consistent with ESD resonance conditions – a value intermediate between our results for homogeneous (60%) and randomly inhomogeneous (20%) plasmas. This difference can be explained by the fact that space waveforms are recorded under a wide range of solar wind conditions, spanning both relatively homogeneous and more strongly inhomogeneous plasmas. Indeed, spacecraft records frequently exhibit isolated wavepackets, suggesting the presence of plasma density fluctuations. This interpretation is further supported by the polarization ratios reported by Graham & Cairns (2013) and Malaspina et al. (2011), which display a broad range of values consistent with variable levels of plasma random inhomogeneities (Polanco-Rodríguez et al. 2025b). Moreover, Graham & Cairns (2013) and Kellogg et al. (2013) found events with two ESD cascades but noted that spectra featuring a larger number of cascades are extremely rare, occurring only in the presence of unusually intense ℒ𝒵 waves. Our results support this conclusion, and we demonstrate that, although not frequent, such ESD cascades can indeed occur.

This paper establishes a direct correspondence between PIC simulation and spacecraft-observed waveforms during type III solar radio bursts. By analysing waveforms recorded under solar wind conditions, this work offers methodological guidance for reproducing and extending these results using Solar Orbiter and Parker Solar Probe spacecraft data. Indeed, decay of an ℒ𝒵 wavepacket to electromagnetic 𝒵 mode was shown with a significant amount of evidence by Polanco-Rodríguez et al. (2026). In this study, direct comparison with waveforms recorded in PIC simulations provides strong support for the interpretation of the observed event, highlighting the potential of this methodology for space plasma physics.

Acknowledgments

This work was granted access to the HPC computing and storage resources under the allocation 2023-A0130510106 and 2024-A017051010 made by GENCI. This research was also financed in part by the French National Research Agency (ANR) under the project ANR-23-CE30-0049-01. C.K. thanks the International Space Science Institute (ISSI) in Bern through ISSI International Team project No. 557, Beam-Plasma Interaction in the Solar Wind and the Generation of Type III Radio Bursts. C. K. thanks the Institut Universitaire de France (IUF). This work has benefited from a grant managed by the Agence Nationale de la Recherche (ANR), as part of the program “Investissements d’Avenir” under the reference ANR-18-EURE-0014.

References

All Tables

All Figures

Fig. 1. Waveforms in a homogeneous and unmagnetized plasma (ΔN = 0, ωc = 0). (a) Time variations in the parallel (grey) and perpendicular (red) electric fields E∥(t) and E⊥(t). (b) Spectral electric field energy |E|2 as a function of the normalized Doppler-shifted frequency ω/ωp and the time ωpt. (c) Time variations in the ion density perturbation δni(t)/n0. (d) Low-frequency spectral energy |δni/n0|2 in the map (ω/ωp, ωpt). (e) High-frequency wave energy spectra |E∥|2 (black) and |E⊥|2 (red), calculated in the time interval 1000 ≲ ωpt ≲ 6000, as a function of ω/ωp. (f) Low-frequency wave energy spectrum |δni/n0|2, in the same time interval and in linear scale, as a function of ω/ωp. (g) Corresponding squared cross-bicoherence $b_c^2$ calculated in the time interval 1000 ≲ ωpt ≲ 6000 for the triplet (E∥, δni, E∥), in the map (ωE∥, ωδni); the extrema bc ≃ 0.72 and bc ≃ 0.68, represented by stars, appear at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })=(0.978,0.0497){\omega }_p$ (first cascade) and $({\omega }_{{\mathcal{L}}^{″}},{\omega }_{{\mathcal{S}}^{″}})=(0.986,0.042){\omega }_p$ (second cascade), respectively. All variables are in arbitrary units.

In the text

Fig. 2. High- and low-frequency energy spectra averaged over N = 256 waveforms, as a function of the normalized Doppler-shifted frequency ω/ωp. (a–b) Parallel (black) and perpendicular (red) averaged electric field spectra ⟨|E∥|2⟩ and ⟨|E⊥|2⟩, in the time intervals 1000 ≲ ωpt ≲ 6000 (a) and 6000 ≲ ωpt ≲ 15 000 (b). (c–d) Low-frequency energy spectra ⟨|δni/n0|2⟩, in the same time intervals as (a) and (b), respectively. (a–b): Logarithmic scales. (c–d): Linear scales. All variables are in arbitrary units.

In the text

Fig. 3. (a) Wave frequency distribution in the map ($\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|/{\omega }_p$, ${\omega }_{\mathcal{S}\prime }/{\omega }_p$), obtained by using the 60% of the spectra that are consistent with ESD occurrence (3 peaks identified), out of a set of 256; the dashed line represents the three-wave resonance condition ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{\mathcal{S}\prime }$. (b) Squared cross-bicoherence ⟨bc⟩2 of the triplet (E∥, δni, E∥), in the time interval 1000 ≲ ωpt ≲ 6000, averaged over the 20% of waveforms (out of 256) satisfying the frequency resonance conditions and high phase coherence in a large (${\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime }$) region. (c) Zoom of (b) in the regions 0.95 < ωℒ/ωp < 1.1 and $0.03<{\omega }_{\mathcal{S}\prime }/{\omega }_p<0.1$; the maximum ⟨bc⟩≃0.67 (⟨bc⟩2 = 0.45) is located at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })\simeq (0.98,0.055){\omega }_p$; the dashed line represents the theoretical curve $({\omega }_{\mathcal{L}}(k),{\omega }_{{\mathcal{S}}^{\prime }}(2k-k_0))$ derived using the ESD resonance condition and the wave dispersion relations. All variables are normalized.

In the text

Fig. 4. Waveforms of the parallel and perpendicular electric fields E∥(t) (grey) and E⊥(t) (red), respectively, to which the time variations in the normalized ion density perturbation δni(t)/n0 are superimposed in (b)–(c) (green lines and right axes), for a plasma with different average levels of density fluctuations: ΔN = 0 (a), ΔN = 0.025 (b), and ΔN = 0.05 (c). Electric fields are in arbitrary units.

In the text

Fig. 5. Waveforms in a randomly inhomogeneous plasma with ΔN = 0.025 and ωc = 0. (a) Time variations in the parallel (grey) and perpendicular (red) electric fields E∥(t) and E⊥(t) (left axis), as well as of the superimposed ion density perturbation δni(t)/n0 (blue, right axis). (b) Spectral electric field energy |E|2 in the map (ωpt, ω/ωp). (c) Time variation in the induced ion density perturbation $\delta {\stackrel{\sim }{n}}_i(t)/n_0$ (applied density fluctuations δn(t) have been removed from δni(t) by filtering). (d) Low-frequency spectral energy $|\delta {\stackrel{\sim }{n}}_i/n_0{|}^2$ in the map (ωpt, ω/ωp). (e) Spectral magnetic field energy |B⊥2|2 in the map (ωpt, ω/ωp). (f) High-frequency wave energy spectra |E∥|2 (black) and |E⊥|2 (red) versus ω/ωp, calculated in the time interval $\mathrm{\Delta }T=[1000,5500]{\omega }_p^{-1}$; green labels and vertical lines indicate the Langmuir spectral peaks. (g) Low-frequency wave energy spectrum $|\delta {\stackrel{\sim }{n}}_i/n_0{|}^2$ versus ω/ωp, calculated within ΔT; green labels indicate the ion acoustic waves spectral peaks. (h) Magnetic wave energy spectrum |B⊥2|2 versus ω/ωp, calculated within ΔT. (i) Squared cross-bicoherence $b_c^2$ calculated within ΔT for the triplet $(E_{\Vert },\delta {\stackrel{\sim }{n}}_i,E_{\Vert })$, in the map (${\omega }_{E_{\Vert }},{\omega }_{\delta {\stackrel{\sim }{n}}_i}$)/ωp; bc ≃ 0.69 at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })=(0.975,0.058){\omega }_p$ (first cascade); bc ≃ 0.85 at $({\omega }_{\mathcal{L}″},{\omega }_{\mathcal{S}″})=(0.99,0.047){\omega }_p$ (second cascade); bc ≃ 0.63 at $({\omega }_{\mathcal{L}″},{\omega }_{{\mathcal{S}}^{(3)}})=(0.99,0.033){\omega }_p$ (third cascade); positions in the frequency map are indicated by stars. (j) Squared cross-bicoherence $b_c^2$ calculated within ΔT for the triplet $(B_{\perp 2},\delta {\stackrel{\sim }{n}}_i,E_{\Vert })$, in the map (${\omega }_{B_{\perp 2}},{\omega }_{\delta {\stackrel{\sim }{n}}_i}$)/ωp; bc ≃ 0.85 at (ωℱ, ω𝒮ℱ) = (1.01, 0.04)ωp, as indicated by stars. Parameters are the same as in Figure 1, but with ΔN = 0.025. All variables are in arbitrary units.

In the text

Fig. 6. (a–b) PDFs of wave frequencies in the plane ($\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|/{\omega }_p$, ${\omega }_{\mathcal{S}\prime }/{\omega }_p$), obtained using the 20% of spectra that show a double-peak structure and satisfy the frequency resonance condition, out of a set of 256, for ΔN = 0.025 and ΔN = 0.05, respectively; the dashed lines represent the three-wave resonance condition ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{{\mathcal{S}}^{\prime }}$. (c–d) Squared average cross-bicoherence ⟨bc⟩2 of the triplet ($E_{\Vert },\delta {\stackrel{\sim }{n}}_i,E_{\Vert }$), calculated in the time interval 1000 ≲ ωpt ≲ 6000, in the frequency map $({\omega }_{\mathcal{L}}/{\omega }_p,{\omega }_{\mathcal{S}\prime }/{\omega }_p)$, for ΔN = 0.025 (⟨bc⟩≃0.54 at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })\simeq (0.98,0.055){\omega }_p$) and ΔN = 0.05 (⟨bc⟩≃0.6 at $({\omega }_{\mathcal{L}},{\omega }_{\mathcal{S}\prime })\simeq (0.97,0.065){\omega }_p$), respectively. Averaging is done over the 8% (c) and 5% (d) of spectra with waves satisfying resonance conditions at frequencies corresponding to high levels of cross-bicoherence, out of 256. The dashed lines represent the parametric curves $({\omega }_{\mathcal{L}}(k)+\alpha \mathrm{\Delta }N{\omega }_p/2,{\omega }_{\mathcal{S}\prime }(2k-k_0))$ for α = −1, 1, derived using the resonance condition ${\omega }_{\mathcal{L}}={\omega }_{\mathcal{L}\prime }-{\omega }_{\mathcal{S}\prime }$ and the wave dispersion relations. Simulation parameters are the same as in Figure 1, but with ΔN > 0. All variables are normalized.

In the text

Fig. 7. Energy spectra averaged over 256 waveforms in the time range ΔT, as a function of the Doppler-shifted frequencies ω/ωp, for ΔN = 0 (blue), ΔN = 0.025 (black), and ΔN = 0.05 (red). (a–c) Parallel ⟨|E∥|2⟩ and perpendicular ⟨|E⊥|2⟩ electric field spectra, as well as ion acoustic energy spectrum $\langle |\delta {\stackrel{\sim }{n}}_i/n_0{|}^2\rangle$, for ΔT = [1000, 6000]ωp−1. (d–f) ⟨|E∥|2⟩, ⟨|E⊥|2⟩, and $\langle |\delta {\stackrel{\sim }{n}}_i/n_0{|}^2\rangle$, for ΔT = [6000, 15 000]ωp−1. All variables are in arbitrary units.

In the text

Fig. 8. Waveforms in a homogeneous and weakly magnetized plasma with ωc/ωp = 0.07. (a) Time variations in the parallel (grey) and perpendicular (red) electric fields E∥(t) and E⊥(t). (b) Spectral electric field energy |E|2 in the map (ωpt, ω/ωp). (c) Time variation in the ion density perturbations δni(t)/n0. (d) Low-frequency spectral energy |δni/n0|2 in the map (ωpt, ω/ωp). (e) Spectral magnetic field energy |B|2 in the map (ωpt, ω/ωp). (f) High-frequency wave energy spectra |E∥|2 (black) and |E⊥|2 (red), versus ω/ωp, calculated in the time interval $\mathrm{\Delta }T=[0,6500]{\omega }_p^{-1}$; the green vertical lines indicate the spectral peaks of excited ℒ𝒵 waves. (g) Low-frequency wave energy spectrum |δni/n0|2 versus ω/ωp, calculated within ΔT; the green vertical lines indicate the spectral peaks of excited ion acoustic waves. (h) Squared cross-bicoherence $b_c^2$ calculated within ΔT for the triplet (E∥, δni, E∥), in the map (ωE∥/ωp, ωδni/ωp); bc ≃ 0.6 at $({\omega }_{\mathcal{LZ}},{\omega }_{\mathcal{S}\prime })=(0.979,0.059){\omega }_p$ (first cascade); bc ≃ 0.63 at $({\omega }_{\mathcal{LZ}″},{\omega }_{\mathcal{S}″})=(0.987,0.048){\omega }_p$ (second cascade); positions are indicated in the map by stars. All variables are in arbitrary units.

In the text

Fig. 9. (a–b) Wave frequency distributions in the plane ($\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|/{\omega }_p$, ${\omega }_{\mathcal{S}\prime }/{\omega }_p$), for Ns selected spectra out of a set of N = 256; (a) : Ns/N≃60%, ωc/ωp = 0.005; (b) : Ns/N≃58%, ωc/ωp = 0.07; both distributions are calculated within the time interval ΔT = [1000, 6000]ωp−1. The dashed lines represent the ESD resonance condition $\left|\mathrm{\Delta }{\omega }_{\mathcal{LL}\prime }\right|={\omega }_{\mathcal{LZ}\prime }-{\omega }_{\mathcal{LZ}}={\omega }_{\mathcal{S}\prime }$. (c–d) Square cross-bicoherence ⟨bc⟩2 averaged over Ns selected spectra with both high bc and ESD frequency resonance conditions, within ΔT, in the map (${\omega }_{\mathcal{LZ}}/{\omega }_p,{\omega }_{\mathcal{S}\prime }/{\omega }_p$), for the triplet (E∥, δni, E∥): Ns/N ≃ 23% (c) and Ns/N ≃ 25% (d). (e–f) Square cross-bicoherence ⟨bc⟩2 averaged over Ns selected spectra, within ΔT, in the map (${\omega }_{\mathcal{LZ}}/{\omega }_p,{\omega }_{\mathcal{S}\prime }/{\omega }_p$), for the triplet (B∥, δni, E∥) : Ns/N ≃ 3.5% (e) and Ns/N ≃ 13% (f). (c–f) : the dashed lines represent the theoretical curves described in Figure 3. (Upper row) : ωc/ωp = 0.005. (Bottom row) : ωc/ωp = 0.07.

In the text

Fig. 10. Energy spectra averaged over N = 256 waveforms, as a function of ω/ωp, for different magnetization ratios ωc/ωp = 0, 0.02, 0.07, and 0.14 (see the legend), in the time ranges 1000 ≲ ωpt ≲ 6000 (top row) and 6000 ≲ ωpt ≲ 15 000 (bottom row). (a,d) Parallel electric energy ⟨|E∥|2⟩. (b,e) Perpendicular electric energy ⟨|E⊥|2⟩. (c,f) Ion acoustic wave spectrum ⟨|δni/n0|2⟩. (a–b, d–e): Logarithmic scales. (c,f): Linear scales. All variables are in arbitrary units.

In the text